Hadamard product of convex functions and Jackson operator

J. Sok\'o\l; K. Piejko; K. Tr\c{a}bka-Wi\c{e}c\a{l}aw

arxiv: 2605.18412 · v1 · pith:4MHXH6PFnew · submitted 2026-05-18 · 🧮 math.CV

Hadamard product of convex functions and Jackson operator

K. Piejko , J. Sok\'o\l , K. Tr\c{a}bka-Wi\c{e}c\a{l}aw This is my paper

Pith reviewed 2026-05-19 23:25 UTC · model grok-4.3

classification 🧮 math.CV MSC 30C45

keywords Jackson operatorHadamard productconvex univalent functionsq-difference operatorunit diskcomplex qgeometric function theory

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The pith

Jackson's difference operator for convex univalent functions equals the Hadamard product of two power series even for complex q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Jackson's q-difference operator, originally defined for real q in the unit disk, can be expressed as the Hadamard product of two power series when applied to convex univalent analytic functions, and that this representation continues to hold for a complex parameter q. This approach unifies the difference operator with coefficient-wise multiplication of series. A reader would care because it provides a concrete algebraic tool for handling q-analogues in geometric function theory, potentially easing the derivation of coefficient bounds or preservation properties under the operator. The work focuses on convex functions that map the unit disk onto convex domains and extends the 1908 real-parameter case to complex values while keeping the operator well-defined.

Core claim

For a convex univalent function f analytic in the unit disk, the Jackson difference operator d_q f(z) with complex q equals the Hadamard product of the power series of f and a second series whose coefficients depend on q, thereby allowing properties of the operator to be read off from the product structure.

What carries the argument

The Hadamard product of two power series, which forms a new series by multiplying corresponding coefficients term by term, used here to rewrite the action of the Jackson q-difference operator.

Load-bearing premise

The functions are analytic convex and univalent in the open unit disk and the complex parameter q is chosen so the difference operator stays well-defined and the Hadamard product representation holds.

What would settle it

Compute the coefficients of d_q f for the identity function f(z) = z with a specific complex q and check whether they exactly match the term-by-term product of the series for f and the q-derived series.

read the original abstract

In this paper we consider some properties of Jackson's difference operator for convex univalent functions in $|z|<1$ with complex parameter $q$ as a Hadamard product of two power series. Jackson in 1908 introduced for a real $q$, $q\in[0,1)$, the difference operator \mbox{${\rm d}_qf(z)$} for an analytic function $f$ in the unit disc $|z|<1$ in the complex plane. Thanks to this operator, many mathematicians have extended the theory of functions in $q$-theory. The $q$-theory has found many applications in theory of hypergeometric series, special functions, combinatorics, number theory, fluid mechanics, quantum mechanics and physics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper writes the Jackson q-difference operator as a Hadamard product on convex univalent functions and extends the parameter q to complex values, but the core identity is algebraic and holds for any power series.

read the letter

The main point is that the authors express Jackson's difference operator for analytic functions in the unit disk as a Hadamard product with the series whose coefficients are (1-q^n)/(1-q). They then study this for convex univalent functions when q is allowed to be complex rather than restricted to the real interval [0,1). This builds on the 1908 operator and the later q-calculus work in geometric function theory. The representation itself comes from matching coefficients term by term, which is direct and works inside the disk of convergence. The paper appears to use this to derive further properties specific to the convex case. That is the extension they are offering. The algebraic identity does not require convexity or univalence, so those conditions are presumably there to obtain the additional results on the functions themselves. The move to complex q is the clearest step beyond the real-q literature mentioned in the abstract. The work stays inside the standard setting of univalent function theory and does not claim to solve larger open problems or produce new applications outside that area. A soft spot is that the Hadamard product step is general and does not depend on the geometric assumptions, so the real content sits in whatever specific properties or bounds they prove afterward. Without seeing the full derivations it is difficult to judge how much is genuinely new versus routine extension. The citation pattern looks standard for the field and does not appear circular. This paper is aimed at researchers already working on q-analogs in complex analysis and geometric function theory. A reader who follows papers on q-univalent functions or Jackson operators would find the complex-q case worth checking. It is modest in scope but formally grounded enough to merit referee time rather than a desk rejection. I would send it for peer review to verify the claimed properties and see whether the complex-q results add anything substantive beyond the real case.

Referee Report

1 major / 2 minor

Summary. The paper considers properties of Jackson's difference operator for convex univalent functions in |z|<1 with complex parameter q, representing the operator as a Hadamard product of two power series. It recalls Jackson's 1908 real-q operator and discusses extensions within q-theory and its applications.

Significance. The algebraic identity expressing the Jackson operator via Hadamard product holds for arbitrary analytic functions by coefficient-wise multiplication and does not require convexity or univalence. Any significance therefore rests on the additional properties (e.g., coefficient estimates or distortion results) that the manuscript derives specifically for convex univalent functions under this representation. If those later results are new and non-trivial, the work could add to q-analogues in geometric function theory.

major comments (1)

[Introduction / main representation theorem] The central representation z D_q f(z) = f * ϕ_q with ϕ_q(z) = ∑ [(1-q^n)/(1-q)] z^n is presented in the context of convex univalent functions, yet the identity follows immediately from the series definition of the operator and holds identically for any power series inside its disk of convergence. Convexity and univalence are not used in establishing the representation itself. Please clarify in the main theorem or §2 which subsequent property genuinely requires these assumptions and would fail for general analytic functions.

minor comments (2)

State explicitly the precise definition of the Jackson operator D_q for complex q (including the normalization factor) and the condition |q|<1 needed for the operator to map the unit disk into itself.
Add citations to prior work on q-univalent functions and Hadamard-product techniques in geometric function theory to situate the contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment point by point below and will make the requested clarifications in the revised version.

read point-by-point responses

Referee: The central representation z D_q f(z) = f * ϕ_q with ϕ_q(z) = ∑ [(1-q^n)/(1-q)] z^n is presented in the context of convex univalent functions, yet the identity follows immediately from the series definition of the operator and holds identically for any power series inside its disk of convergence. Convexity and univalence are not used in establishing the representation itself. Please clarify in the main theorem or §2 which subsequent property genuinely requires these assumptions and would fail for general analytic functions.

Authors: We agree that the identity z D_q f(z) = (f ∗ ϕ_q)(z) is purely algebraic and holds for any analytic function f inside its disk of convergence by direct coefficient comparison; convexity and univalence play no role in its derivation. The manuscript places this representation in the setting of convex univalent functions because the subsequent results—sharp coefficient estimates, distortion theorems, and preservation properties under the Jackson operator—rely on the growth and coefficient bounds that are specific to the class of convex univalent functions and do not hold for arbitrary analytic functions. In the revised manuscript we will restate the main theorem in §2 so that the general identity is presented first, followed by an explicit indication of the additional assumptions required for the quantitative results that follow. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algebraic identity independent of convexity assumptions

full rationale

The central representation is the algebraic identity z D_q f(z) = f * ϕ_q, which follows termwise from the definition of the Jackson operator and the Hadamard product for any power series inside the disk of convergence. Convexity and univalence are invoked only for subsequent properties, not to establish the representation itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The result is self-contained against the 1908 Jackson operator and standard q-calculus.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Paper relies on standard domain assumptions from complex analysis and q-calculus; no free parameters, new entities, or ad-hoc axioms are identifiable from the abstract alone.

axioms (2)

domain assumption The functions are analytic and convex univalent in the unit disk |z|<1
Standard setting for the class of functions studied in geometric function theory
domain assumption The Jackson operator with complex q is well-defined for the given functions
Required for the difference operator to make sense in the complex case

pith-pipeline@v0.9.0 · 5668 in / 1172 out tokens · 31724 ms · 2026-05-19T23:25:26.397572+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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[2]

F. H. Jackson, Onq-functions and certain difference operator, Trans. Royal Soc. Edinburgh 46(1908) 253–281

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F. H. Jackson, Onq-definite integrals, Quarterly J. Pure and Appl. Math. 41(1910) 193–203

work page 1910
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Piejko, J

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Piejko, J

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M. S. Robertson, On the theory of univalent functions, Ann. Math. (2)37(1936) 379–408

work page 1936
[8]

M. S. Robertson, Certain classes of starlike functions, Michigan Math. J. 32(1985) 135–140

work page 1985
[9]

Ruscheweyh, T

St. Ruscheweyh, T. Sheil–Small, Hadamard product of schlicht functions and the P` olya-Schoenberg conjecture, Comm. Math. Helv. 48(1973) 119–135

work page 1973
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Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans

S. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38(1935) 310–340. 1 Rzesz´ow University of Technology, Faculty of Mathematics and Applied Physics, Al. Pow- sta´nc´ow Warszawy 12, 35-959 Rzesz´ow, Poland Email address:piejko@prz.edu.pl 2 University of Rzesz ´ow, ul. Prof. Pigonia 1, 35-310 Rzesz ´ow, ...

work page 1935

[1] [1]

A. W. Goodman, Univalent Functions, II, Mariner Publishing Co.: Tampa, Florida (1983)

work page 1983

[2] [2]

F. H. Jackson, Onq-functions and certain difference operator, Trans. Royal Soc. Edinburgh 46(1908) 253–281

work page 1908

[3] [3]

F. H. Jackson, Onq-definite integrals, Quarterly J. Pure and Appl. Math. 41(1910) 193–203

work page 1910

[4] [4]

Noshiro, On the theory of schlicht functions, J

K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. Jap., 2(1)(1934-35) 129–135

work page 1934

[5] [5]

Piejko, J

K. Piejko, J. Sok´ o l, On convolution andq-calculus, Boletin Soc. Mat. Mexicana, 26(2)(2020) 349–359

work page 2020

[6] [6]

Piejko, J

K. Piejko, J. Sok´ o l, K. Tr¸ abka-Wi¸ ec law, Onq-calculus and starlike functions, Iran. J. Sci. Technol. Trans. A Sci. 43(6)(2019) 2879–2883

work page 2019

[7] [7]

M. S. Robertson, On the theory of univalent functions, Ann. Math. (2)37(1936) 379–408

work page 1936

[8] [8]

M. S. Robertson, Certain classes of starlike functions, Michigan Math. J. 32(1985) 135–140

work page 1985

[9] [9]

Ruscheweyh, T

St. Ruscheweyh, T. Sheil–Small, Hadamard product of schlicht functions and the P` olya-Schoenberg conjecture, Comm. Math. Helv. 48(1973) 119–135

work page 1973

[10] [10]

Strrohh¨ acker, Beitrage z¨ ur Theorie der schlichter Functionen, Math

E. Strrohh¨ acker, Beitrage z¨ ur Theorie der schlichter Functionen, Math. Z. 37(1933) 356–380

work page 1933

[11] [11]

Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans

S. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38(1935) 310–340. 1 Rzesz´ow University of Technology, Faculty of Mathematics and Applied Physics, Al. Pow- sta´nc´ow Warszawy 12, 35-959 Rzesz´ow, Poland Email address:piejko@prz.edu.pl 2 University of Rzesz ´ow, ul. Prof. Pigonia 1, 35-310 Rzesz ´ow, ...

work page 1935